The Stacks project

Lemma 10.155.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $\kappa \subset \kappa ^{sep}$ be a separable algebraic closure. There exists a commutative diagram

\[ \xymatrix{ \kappa \ar[r] & \kappa \ar[r] & \kappa ^{sep} \\ R \ar[r] \ar[u] & R^ h \ar[r] \ar[u] & R^{sh} \ar[u] } \]

with the following properties

  1. the map $R^ h \to R^{sh}$ is local

  2. $R^{sh}$ is strictly henselian,

  3. $R^{sh}$ is a filtered colimit of ├ętale $R$-algebras,

  4. $\mathfrak m R^{sh}$ is the maximal ideal of $R^{sh}$, and

  5. $\kappa ^{sep} = R^{sh}/\mathfrak m R^{sh}$.

Proof. This is proved by exactly the same proof as used for Lemma 10.155.1. The only difference is that, instead of pairs, one uses triples $(S, \mathfrak q, \alpha )$ where $R \to S$ ├ętale, $\mathfrak q$ is a prime of $S$ lying over $\mathfrak m$, and $\alpha : \kappa (\mathfrak q) \to \kappa ^{sep}$ is an embedding of extensions of $\kappa $. $\square$

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